Multidimensional method and system for statistical process control

ABSTRACT

The present invention relates to a method and to any system using such a method for performing statistical process control on the basis of taking indicators or measurements on the inputs, the outputs, and the control and operating parameters of said process, and which can be represented by observation points in frames of reference associating their values to their sampling indices. In the invention, the observed values are transformed so that the resulting values are compatible with the multidimensional Gaussian distribution model; said transformed observation points are situated in a multidimensional space, each dimension being associated with a measured magnitude; the out-of-control observation points situated in this way and concentrated in a particular direction are identified; and said direction is associated with a common cause for drift in said process, and each observation-and-anomaly pair is associated with indicators in order to propose zero, one, or more causes of anomaly that might relate to the observation made.

This is a U.S. national stage of application No. PCT/FR99/02254, filedon Sep. 22, 1999. Priority is claimed on that application and on thefollowing application Country: France, application No.: 98/12113, filedon Sep. 22, 1998.

The present invention relates to a method and any system using such amethod of statistical process control based on multidimensionalprocessing of data.

Its purpose is firstly to trigger a warning when the process departsfrom “normal” operation which ensures that its production is of therequired quality, and secondly to make proposals for identifying theprobable cause(s) of the anomaly.

Statistical process control (SPC) is presently in use in a very largenumber of businesses, in all countries (mainly industrializedcountries), for all types of industrial production: engineering,electronics, chemistry, pharmaceuticals, agri-food, plastics materials,. . . .

Its purpose is to ensure product quality by inspecting the manufacturingprocess itself and not only by inspecting the characteristics of itsproducts. SPC has become essential in achieving “zero defects” and whenthe business seeks to comply with international quality assurancestandards (ISO 9000).

Its technical objective is to detect possible drift in the manufacturingprocess and to remedy it before non-compliant products are manufactured.

The use of this method has now extended beyond the context ofmanufacturing goods and covers producing services (banking, insurance,consultancy, . . . ).

When running a process (cf. FIG. 1), various measurements (indicators)associated with the same process are tracked: input characteristics (rawmaterials); output characteristics (products); process operatingparameters. Each unit of observation (measurement instant or elementproduced) is thus associated with a plurality of digital values obtainedby the measurements, thus enabling it to be represented by a point inthe multidimensional space of the measurements taken.

The usual practice in SPC consists in monitoring the process by trackinga plurality of control charts which are graphical representations of theway an observed magnitude varies and which present predefined controllimits (see FIG. 2), one per measurement. Each control chart is theninterpreted independently of the others, triggering warningsindependently.

Various types of control chart are available (known as “Shewart, CuSum,EWMA, MME”), with the last three being accepted as being better atdetecting small amounts of “drift” than the first.

Generally, control charts are used on grouped data: by plotting theaverages of a plurality of grouped-together measurements, small amountsof drift are detected better, and in addition the distribution of thevalues coincides better with the assumption of normality that underliesthe method. By plotting the variances or the extents of each group, itis possible to detect an increase in measurement variability that hassome special cause.

The usual practice which consists in simultaneously and independentlymonitoring a plurality of control charts constitutes a method that isclumsy and not very effective in multidimensional SPC:

it raises too many false warnings which can give rise to unnecessarycorrections; these then need to be reassessed very quickly, and lead tothe process being controlled in a manner that is chaotic and expensive,with multiple corrections;

it can detect real anomalies too late; and

it has difficulty in detecting the causes of anomalies when they are notdirectly associated with a measurement. This encourages taking amultiplicity of measurements which is expensive and leads to amultiplicity of control charts.

The method and system of the present invention seek to mitigate thosedrawbacks: the method is one of statistical process control on the basisof taking indicators or measurements on inputs, on outputs, and oncontrol and operating parameters of said process, and which can berepresented by observation points in frames of reference that associatetheir values with their sampling indices; according to the invention:

a) the observed values are transformed so that the resulting values arecompatible with the multidimensional Gaussian distribution model, andconstitute data corresponding to the observation points used in theremainder of the method;

b) said observation points are situated in a multidimensional space, inwhich each dimension is associated with a measured magnitude;

c) amongst the observation points, points that are said to be “undercontrol” and that correspond to proper operation of the process aredistinguished from points which are said to be “out of control”;

d) the distribution center of the points under control is calculated asbeing the center of gravity of the observation points under control;

e) out of control observation points that are concentrated in someparticular direction from the distribution center of the points undercontrol are identified;

f) this direction is associated with a common cause for drift of saidprocess;

g) each observation point and anomaly direction pair is associated withindicators in order to propose zero, one, or more causes of anomaly thatare liable to be associated with the observation that has been made; and

h) when an anomaly is analyzed in this way, a warning is triggered andthe drift detected in this way in the industrial process is remedied.

Said center of gravity of the observation points being inspectedcorresponds to a point whose components are the means of the componentsof the observation points under inspection.

SPC inspection consists in conventional manner in regularly observing pcontinuous magnitudes y¹, y², . . . , y^(p) either statistically or bysampling. These magnitudes can equally well represent characteristics ofraw materials, characteristics of manufactured products, or operatingparameters of the manufacturing process. The p-dimensional vector madeup of these p measurements at a given “instant” is written y and isreferred to as the observation vector of the process, with the endpointof this vector being the observation point of the process and the originof this vector being the original of the frame of reference in question.

It is clear that in this context the concept of “instant” goes beyond astrictly temporal interpretation: measurements associated with the same“instant” are, wherever possible, measurement of parameters relating tothe production of the same manufactured unit or batch. Perfecttraceability of the manufacturing process is then necessary in order tobe able to define which measurements are associated with the same“instant”.

When the process is “under control”, the values of y at varioussuccessive instants t₀, t₀+1, t₀+2, . . . vary “little” about a valuey0, which is the desired target for ensuring that production is ofsatisfactory quality. This variation is due to random variations in thecharacteristics of the raw materials (material hardness, chemicalcomposition of a component, supplier, . . . ), of the environment(temperature, humidity, . . . ), or of the process (setting of amachine, attention of an operator, . . . ). These characteristics haveinfluence over one or more components of y and they are written z¹, z₂,. . . , z^(m) and together they form a vector written z. The vector z isreferred to herein as the explanatory vector of the process.

For a characteristic of the process to be considered as an “observationvariable” of the process y^(j) it must be evaluated at each “instant”.

For a characteristic of the process or of the inputs to be considered asa “cause variable” z^(k) of the process, it must be modified by an agentexternal to the system proper: voluntary or involuntary human action,variation in the environment, wear, or aging. Generally, for reasons ofexpense or of feasibility, these variables are not measured at each“instant” (otherwise they would also appear as variable y^(j)) and inthis sense they constitute “hidden variables” that influence thebehavior of the process. Evaluating them is often expensive, lengthy,imprecise, and is performed only in the event of an anomaly.

A variable can be quantitative if its possible values are numerical andbelong to a known range of values (temperature, pressure, . . . ), or itcan be qualitative when the possible values, numerical or otherwise, arelimited in number (supplier, operator, machine, . . . ). The models andmethods considered in the present invention assume that the componentsof y are all quantitative.

The same characteristic of the process (e.g. the controlled temperatureof a furnace) can appear both as a component of z as a component of y.

The dependency between y and z can be modelled by the followingrelationship:

y=f(z,t)+ε

where t designates the observation instant and ε is a random vector ofdimension p whose average is assumed to be zero and which has acovariance matrix Σ_(ε). f is a vector function having p components f¹,f², . . . , f^(p) such that y_(i)=f^(j)(z,t)+ε.

The components of y are correlated with one another as are thecomponents of z.

A perfectly stabilized process under steady conditions ought to presentthe following aspects:

f(z,t) does not in fact depend on t;

each “cause variable” z^(k) is stabilized on a fixed value z₀ ^(k); and

y can be modelled by a steady process of the form:

y=f(z ₀)+ε

In reality, it is not possible to determine perfectly the quantitativecause variables: the variable z₀ has added thereto a random error ofzero expectation and of covariance matrix Σ_(e). The model then becomes:

z ^(k) =z ₀ ^(k) +e ^(k)

y=f(z ₀ +e)+ε

The method proposed by the invention lies in the following context whichis the usual context for SPC:

the function f is unknown;

the explanatory variables z^(k) are not all identified; and

n observations have been made of the variables y¹, . . . , y^(p) at the“instants t=1, . . . , t=n.

These observations are written in the form of a matrix Y having n rowsand p columns. y^(j) designates the jth column of Y; the ith element ofthis column is written y_(i) ^(j) and designates the observation made atinstant t=i of the variable y^(j). The vector of observations of thevariables y¹, . . . , y^(p) at the “instant” t=i is written y_(i).

The observation y_(i) is given a weight p_(i), which is generally equalto 1/n. The diagonal matrix (n, n) having the weights p_(i) as diagonalterms is written D_(p).

when the process is properly under control:

it is properly centered on the target value y0.

y0 is then equal to the mathematical expectation E[y] of y; y0 is thusvery close to the observed mean value m_(y) (where m_(y) designates thevector constituted by the means {overscore (y)}¹, . . . , {overscore(y)}^(p) of the p columns y¹, . . . , y^(p) of the matrix Y).

its variability is constant and comparable with the specification limitsdefined on the various variables y^(j),

the covariance matrix of the random vector y does not vary in time.

when the process is drifting, the observed values y_(i) move too faraway from the target value y0. Such behavior can be the result of:

variation over time in the center value of one or more cause variablesz^(k); or

an increase in time in the variance of one or more randoms ε^(k) ore^(j).

If the drift is qualitative concerning the cause variable z^(k), causingthis variable to go from a value z0 ^(k) to a value z1 ^(k), then thecenter of the distribution is moved from y0 to y1. The observed pointsare then moved in the direction y1-y0.

If the drift is quantitative concerning z^(k) and the process is notunstable around y0 relative to z^(k), then it can be assumed that eachfunction f^(j) has a partial derivative f^(jk) relative to z^(k). Thecalculus of variations then shows that to the first order:

drift in the mean, z1 ^(k)=z0 ^(k)+d implies that the center of theobserved points moves from y0 in the direction defined by the vector ofthe partial derivatives (f^(1k)(y0), . . . , f^(jk)(y0), . . . ,f^(pk)(y0));

an increase in the variability of the random e^(k) will give rise to thedistribution of the observed points y_(i) being “stretched” in the samedirection:

(f ^(1k)(y 0), . . . , f ^(jk)(y 0), . . . , f ^(pk)(y 0))

an increase in the variability of the random e^(j) gives rise to thedistribution of the observed point y_(i) being “stretched” in thedirection of the jth basic vector (0, …  , 0, 1, 0, …  , 0)1  j  p

The method of the present invention seeks and manages to achieve thefollowing technical objectives:

during a historical analysis stage:

identifying the directions associated with any drift that has beenidentified in the historical record and define parameters making itpossible for each observation to calculate proximity indicators forthese directions;

during an operational stage of controlling the process:

detecting whether the latest observation reveals any drift in theoperation of the process and then, by examining the proximityindicators, identifying the identified cause direction that appearsclosest to the observed point, thus proposing a probable cause or causesfor the drift;

proposing in both stages graphical representations that enable thesituation to be evaluated quickly and as a whole.

In order to adapt the system to the particular features of each process,several versions are proposed for each kind of processing performed inthe system.

The present invention provides a method of statistical process controland any system using such a method based on measured indicators ormeasurements of inputs, outputs, and control and operating parameters ofsaid process and comprising various kinds of processing on the valuesobtained in this way: it preferably operates by means of a computerperforming computer processing, and in the processing step operatingentirely automatically or while assisting the user.

Its operation takes place in two stages:

A learning stage including performing historical analysis of theprocess; the history is made up of a set of magnitudes associated withthe operation of the process; these magnitudes are measured or evaluatedat successive instants or on parts taken by sampling.

The purpose of this analysis is:

to identify

the instants at which drift in the process becomes manifest; and

the special cause(s) of the observed drift (by exchanging informationwith a specialist user of the real process under study).

to evaluate the values of the parameters which define the directionsassociated with each identified special cause.

A process-tracking stage.

During this stage, the system receives measurements and data from theprocess (from sensors over direct links, or via manual input). It thenenables warnings to be triggered when the process drifts. The magnitudesassociated with each cause are evaluated for this observation, so italso specifies the probable cause(s) of the drift, which causes areselected from those that have previously been identified by the analysisof the method. If no cause is proposed, then the cause is identified bycausing a human to examine the real process; the newly identified causeis then integrated in the system so as to enable it to be identifiedautomatically should it occur again at a later date.

The pertinence and the originality of the system of the presentinvention come from:

the appropriateness of the model proposed and the extent of itsapplications;

the properties of the trace of an anomaly in the multidimensional spaceof the observed magnitudes: a straight line in the observation spaceR^(n);

the definition of composite magnitudes that are characteristic of eachidentified cause and the associated probability laws: “cause intensity”and “angular proximity”;

the powerful multidimensional statistical analysis methods adapted tothis model that are used: classification of out-of-control points,determining cause directions;

the pertinence of the proposed graphical representations: control chartsgraduated with probability thresholds; and

by analyzing the history of the process, the learning stage contributesto identifying anomalies using the following steps:

A1—Prior transformation of the data;

A2—Distinguishing between under-control measurements and out-of-controlmeasurements;

A3—Identifying types of special causes and associated directionparameters;

A4—Creating and examining control charts on the special causeindicators; and

A5—Interpreting the special causes.

These steps are implemented interactively with specialists in the realprocess who can intervene to introduce new information into the system(observations recognized as being out of control, associating aplurality of causes, identifying measurements that are suspect, . . . )and to take advantage of partial information provided by the system inorder to refine interpretation thereof in the history. Results obtainedin one step often make it possible to repeat one or more earlier stepswith modifications to options or certain parameters.

The history of a process is characterized by the matrix Y having n rowsand p columns. The element y_(i) ^(j) situated at the intersection ofrow i and column j represents the value taken by the variable y^(j)during the observation at “instant” i.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a general depiction of a process of the type capable of beingcontrolled by the present invention;

FIGS. 2A and 2B graphically depict data control charts for two monitoredparameters of a controlled process;

FIG. 3 illustrates an anomaly situation in data observation space usingraw data values;

FIG. 4 illustrates an anomaly situation based on normalized principalcomponents of controlled data;

FIG. 5 graphically depicts a T² control chart with a corresponding P_(u)chart; and

FIG. 6 graphically depicts a cosine control chart with a correspondingP_(b) chart.

Step A1: Prior Transformation of Data

This relates to transforming observed raw data y_(i) ^(j) so that thevalues on which the processing is applied are closer to the modelunderlying the method as described above, thereby increasing thepertinence of the results.

The transformations that are most generally useful are performedseparately on each observed variable, transforming the raw value of anobservation y_(i) ^(j) into a new value {tilde over (y)}_(i) ^(j). Thefollowing transformations can be mentioned, but they are not exclusive:

Successive readings can be grouped together to form a single averagedreading. (Cf. control charts for grouped data.) For example:${\overset{\sim}{y}}_{h}^{j} = {\frac{1}{w}{\sum\limits_{s = 1}^{w}\quad y_{{{({h - 1})} \cdot w} - s}^{j}}}$

where w is the size of each group.

The differences between successive measurements in order to decorrelatemeasurements:

{tilde over (y)} _(i) ^(j) =y _(i) ^(j) −y _(i+1) ^(j)

transformations giving a distribution that is “closer” to Gaussiandistribution. For example:

{tilde over (y)} _(i) ^(j) +F _(G) ⁻¹(F _(y) _(^(j)) (y _(i) ^(j)))

where F_(G) ⁻¹ is the function that is the inverse of the Gaussiandistribution function, F_(y) _(^(j)) is the function assumed for thedistribution of the variable y^(j).

centering on presumed centers of the random variations; in which case:

{tilde over (y)} _(i) ^(j) =y _(i) ^(j) −c _(i) ^(j)

Where the choice of the center value c_(i) ^(j) depends on concretesituations, knowledge about the process. Thus, this can be:

c_(i) ^(j)=the average of under control values of the variable y^(j);

c_(i) ^(j)=the target y0 ^(j) of the variable y^(j), which is constantover time; or

c_(i) ^(j)=the target y0 _(i) ^(j) of the variable y^(j) which is timevariable (small series, known and acceptable variations, . . . ).

scale transformations so as to make the scales of unrelated magnitudesmore uniform or so as to take account of the importance given a priorito the variability in the observed measurements:

{tilde over (y)} _(i) ^(j) =y _(i) ^(j) /s _(i) ^(j)

where s_(i) ^(j) designates the unit selected for this measurement, forexample:

s_(i) ^(j)=the standard deviation y^(j) for under-control observations.

Step A2: Distinguishing Under-control Measurements and Out-of-controlMeasurements

This consists in taking the measurements of the history anddistinguishing between those which do not correspond to normal operationof the process while it is under control.

Initially, any observations which are recognized as being not undercontrol by the people in charge of the process are labeled as being outof control.

Then, on observations that might still be considered as being undercontrol, a Hotelling T² chart is constructed (see the work by Douglas C.Montgomery entitled “Introduction to statistical quality control”,second edition, published by Wiley in 1991). Observations situatedbeyond the upper control limit associated with the threshold α aredeclared to be out of control. α designates the acceptabilityprobability of false warnings. In general, the value selected for α is0.05 or 0.001.

This step needs repeating: on each iteration, new observations arelabeled as being out of control.

Iteration is stopped when the number of observations that lie outsidethe control limits of the T² chart is compatible with the acceptableprobability α of false warnings.

This compatibility is evaluated by a conventional hypothesis test: giventhe observed frequency of out-of-limit values, the hypothesis is testedof having an out-of-limit measurement probability that is less than orequal to α (see the work by G. Saporta, “Probabilité, analyse desdonnées et statistique” [Probability, data analysis, and statistics],published by Technip, 1990).

A3—Identifying the Special Cause Types and the Associated DirectionParameters

For each observation in the history labeled as being out of control inthe preceding step, the idea is to associate a well-identified cause forthe anomaly, referred to as its “special cause”.

This identification must be performed by using the knowledge of thepeople in charge of the process. Nevertheless, a well adapted automaticclassification method applied to points that are recognized as being outof control can guide those people: such classification brings togetherobservations which are capable of corresponding to a common cause of ananomaly, i.e. it brings together observation points that are close to acommon direction in the multidimensional space.

A method that is suitable for this purpose is the automaticclassification method of the hierarchical rising classification typeusing the absolute value of the cosine as a similarity index between twoobservation vectors and accepting the maximum binding criterion, alsoknown as the diameter criterion, for making up groups (see theabove-cited work by G. Saporta). Classification then applies to theout-of-control measurements which are centered on the distributioncenter of the out-of-control data.

An anomaly direction is more easily interpreted if the measurements mostconcerned with the anomaly are determined.

To do this, linear regression is performed without a constant term ofthe “step by step” or “best subset” type, taking the direction of theanomaly as the variable to be explained and the measured variables,after transformation and centering on the under-control distributioncenter as the explanatory variables.

On examining the variables selected in this way and the signs of theresulting regression coefficients, the people in charge of the processwill be guided in determining the original of the anomaly.

A4—Creating and Examining Control Charts on the Indicators of SpecialCauses

At the end of the preceding step, several groups of out-of-controlobservations are available, each group being associated with aparticular anomaly or “special cause”.

This step begins by associating each special cause with its principaldirection whose existence is mentioned above, and then with twoindicator functions. The two indicators are functions which, whenapplied to any observation, indicate whether the observation is capableof coming from abnormal operation of the process having this particularspecial cause.

FIGS. 3 and 4 illustrate as dimension 2 the directions associated withtwo different anomalies, FIG. 3 shows the situation in data observationspace using raw values, while FIG. 4 shows the situation in the frame ofreference based on normalized principal components of under-controldata. It can be seen that the separation between the two types ofanomaly is more marked in FIG. 4 than in FIG. 3.

The observation points that are then taken into consideration are to befound in the frame of reference based on the normalized principalcomponents of measurements under control, so its origin coincides withthe center of distribution of the under-control data.

The theoretical model described above leads to searching for thedirection representative of an anomaly as extending from the center ofdistribution of under-control observations and “close” to the set ofobservation points that are due to the anomaly. This direction can bedefined as the first inertial axis of the observation points associatedwith said anomaly, said axis passing through the origin (cf. thetransformation mentioned above). It is known that such an axis is thefirst principal axis determined by principal component analysis that isnot centered and not reduced to the cloud of observation pointsassociated with the anomaly (see the above-cited work by G. Saporta).

After evaluating the direction in which the points associated with aspecial cause go away from the distribution center of the under-controlpoints, it is possible to define two indicators associated with theanomaly:

the first indicator measures the distance to an observation point insaid direction; and

the second indicator measures the proximity of a point to saidparticular direction in the observation space R^(n), in terms of angularproximity seen from the distribution center of the points that are outof control.

In the multidimensional observation space, the first indicator measuresthe distance of the observation along the direction associated with thespecial cause. It associates each observation-and-anomaly pair with anindicator of the intensity of the relationship between the observationand the anomaly, referred to as the “cause variable”, obtained as thecomponent of the observation point along the direction associated withthe cause, with this indicator being calculated as a scalar product ofthe observation vector with the direction vector for the direction ofthe cause.

By calculating this scalar product in the normalized principal componentspace, that amounts to taking the matrix V⁻¹ in the initial measurementspace as the scalar product matrix, where V⁻¹ designates the inverse ofthe covariance matrix V of the data under control when V is of fullrank, and V⁻¹ designating the pseudo-inverse matrix of V if V is not offull rank.

This indicator is thus a linear combination of initial measurements, andon the assumption that these measurements obey a Gaussian law (when theprocess is under control), this indicator also obeys a Gaussian law; itsvariation can thus be followed using a conventional control chart. Thus,an observation situated out of control on such an intensity chart ishighly likely to correspond to the type of anomaly that is representedby the indicator marked on the chart.

However, when an observation point is very far from the distributioncenter of points under control, it can appear to be far away in severalcause directions and thus to be associated with several types ofanomaly. To lift the ambiguity, the second indicator is then considered.

This is an angular proximity indicator between the directionrepresenting the anomaly and the vector whose origin is at the center ofthe data under control and whose end is at the observation point. Thisangle can be evaluated as its cosine since that is easy to calculate:the cosine is calculated by dividing the above intensity indicator bythe norm of the vector representing the observation.

Since this calculation is performed in normed principal component space,it is more pertinent and it can be used to define control limits thatcan be marked on the control chart relating to this new indicator.

When operation is under control, this indicator has a probabilitydistribution whose distribution function F can be determined.

The function F is used to define the control limit Lα of the angularproximity indicator beyond which an observation is allocated to saidanomaly, said function F representing a distribution function of thecosine of the angle (x) formed by an arbitrary given direction in R^(n),where n≧2, and a centered Gaussian random vector R^(n) having theidentity matrix as its covariance matrix, said distribution function Fof said random variable being given by the following formula:$\begin{matrix}{{{if}\quad x} < {- 1}} & {{F(x)} = 0} \\{{{if}\quad x} > 1} & {{F(x)} = 1} \\{{{if}\quad x} \geq {{- 1}\quad {and}} \leq 1} & {{F(x)} = {1 - \frac{S\left( {{ArcCos}(x)} \right.}{S(\pi)}}}\end{matrix}$

where S(θ) = ∫₀^(θ)sin^(n − 2)(t)  t

such that Lα=F⁻¹(α), where α represents the acceptable probability for afalse warning, α preferably representing a value lying in the range0.001% to 0.05%. The term “false warning” is used herein to mean anobservation that is under control being erroneously allocated to saidanomaly.

By examining the angular proximity control chart it is possible toselect from the points declared to be out of control by the “intensity”chart those points which genuinely depend from the anomaly underconsideration: these points are declared to be out of controlsimultaneously on both charts.

In practice, such an angular proximity chart is not very readable sincewith a small number of dimensions the control limits are very close to 1or −1. That is why it is preferred to use charts representing theseindicators after appropriate transformation has been performed and on achart expressed in probability thresholds and having a scale that islogarithmic.

This mode of representation can be useful for all types of magnitude tobe inspected; it is easier to interpret and provides greater uniformityin presentation for charts associated with different probabilitydistributions. Two types of chart are defined:

the unilateral chart P_(u) which can be used when control applies to asingle limit (as in the case of the T² chart); and

the bilateral chart P_(b) which is used when the control relates to alower limit and an upper limit.

An observation value x associated with the index i of a distributionfunction F_(x) is marked on a unilateral chart P_(u). The observation isthen plotted on the chart at abscissa i and ordinatey=min(log₁₀(1−F_(x)(x), 4).

The ordinate scale is graduated by threshold values associated withintegral values for y: 1−F_(x)(x)=10(^(−y)). Ordinate 4 on the scale isassociated with the mention “<0.0001” to take account of the truncationthat has been performed. A horizontal line marks the selected controllimit. FIG. 5 shows a conventional T² control chart and thecorresponding P_(u) chart.

An observation value x associated with the index i of a distributionfunction F_(x) is marked on a bilateral chart P_(b). The observation isthen marked on the chart at abscissa i and ordinate y as defined by:

if x≦m y=min(−log₁₀((1−F _(x)(x))*2), 4)

if x>m y=min(−log₁₀(F _(x)(x))*2), 4)

where m designates the mean of the distribution: F_(x)(m)=0.5.

The ordinate scale is graduated in threshold values associated withinteger values of y: 10^((−|y|)). A horizontal line marks the controllimit associated with the selected control threshold. FIG. 6 shows aconventional cosine control chart and the corresponding P_(b) chart.

A5—Interpretation of Special Causes

The above-defined control charts give the people in charge of theprocess additional information enabling them to discover all of thepoints in the history which do not correspond to proper operation of theprocess, and for each of them to identify the real cause of the anomaly.The tools described above thus enable each identified cause of ananomaly to be associated with a direction in the observation space andwith two indicators enabling it to be identified.

The second stage of the method of the invention is that of controlproper, and it comprises the following steps:

S1—the data is initially transformed;

S2—anomalies are detected and identified by monitoring using controlcharts; and

S3—where necessary, a combination of causes or a new special cause isidentified and integrated into the method.

During this second stage, the system receives measurements and datacoming from the process (coming from directly connected sensors or inputmanually), which measurements are based on the model of the analyzedhistorical measurements. It then enables warnings to be triggered whenthe process drifts. It then specifies the probable cause(s) of thedrift, where causes are selected amongst those that have already beenidentified during the learning stage.

Each observation which is received from the process is processed inseveral steps:

S1—Prior Transformation of the Data

The data is subjected to the transformations as defined in step A1.

S2—Detecting and Identifying Anomalies

The value of T² associated with the observation using the calculationmodel defined in step A2 is calculated and marked relative to thecontrol limit of the T² chart.

If the chart finds that the observation is under control, then theprocessing of the observation terminates, otherwise the observation isrepresentative of the process not operating properly.

Under such circumstances, the values taken by the indicators associatedwith the various causes identified during the learning stage arecalculated and the position of the point representing the newobservation is examined on P_(b) type control charts associated withsaid indicators. When both indicators associated with the same causeindicate that the observation is out of control, then this cause istaken as possibly being at the origin of actual misfunction of theprocess; the system sends a warning to the process controller concerningthe misfunction together with its diagnosis as to the origin of themisfunction.

Processing of the observation then terminates, unless the system hasalso identified and recognized another one of the listed causes. Undersuch circumstances, the following additional step is necessary.

S3—Where Necessary, a Combination of Causes or a New Special Cause isIdentified and Integrated in the System

When no cause is recognized in the preceding step, then tests are madeto see whether the observed anomaly stems from the simultaneousappearance of two known causes.

For this purpose, linear regression is performed without a constantterm, of the “step by step” type or of the “best subset” type, takingthe observation point centered on the center of the under-controldistribution as the variable to be explained and the cause variables asthe explanatory variables.

If a large multiple correlation coefficient R is obtained while usingonly two explanatory variables, and in particular if R>0.95, then it canbe considered that there is a simultaneous occurrence of the two causesassociated with these two variables. It is possible in like manner toconsider the possibility of more than two causes occurringsimultaneously, in particular three causes or four causes.

When no cause or combination of causes in the list is recognized, thenit is necessary to perform manual identification based on the knowledgeand experience of the people in charge of the process.

What is claimed is:
 1. A statistical method of controlling an industrialprocess on the basis of readings of indicators or measurement ofmagnitudes characteristic of inputs, outputs, and control and operatingparameters of said process, and capable of being represented byobservation points in frames of reference associating their values totheir sampling indices, said indicators or measurements being given bysensors or by manual input, in which: a) the observed values aretransformed so that the resulting values are compatible with themultidimensional Gaussian distribution model, and constitute datacorresponding to the observation points used in the remainder of themethod; b) said observation points are situated in a multidimensionalspace, in which each dimension is associated with a measured magnitude;c) amongst the observation points, points that are said to be “undercontrol” and that correspond to proper operation of the process aredistinguished from points which are said to be “out of control”; d) thedistribution center of the points under control is calculated as beingthe center of gravity of the observation points under control; e) out ofcontrol observation points that are concentrated in some particulardirection from the distribution center of the points under control areidentified; f) this direction is associated with a common cause fordrift of said process; g) each observation point and anomaly directionpair is associated with indicators in order to propose zero, one, ormore causes of anomaly that are liable to be associated with theobservation that has been made; and h) when an anomaly is analyzed inthis way, a warning is triggered and the drift detected in this way inthe industrial process is remedied.
 2. A statistical process controlmethod according to claim 1, in which each observation and anomaly pairis associated with an indicator giving the intensity of the relationshipbetween the observation and the anomaly, referred to as the “causevariable”, the indicator being obtained as the component of theobservation point along the direction associated with the cause, saidindicator being calculated as a scalar product of the observation vectormultiplied by the direction vector for the direction of the cause, thescalar product matrix used as an indicator of the intensity of therelationship between an observation and an anomaly being the matrix V⁻¹where V⁻¹ designates the matrix which is either the inverse of thecovariance matrix V of the data under control when V is of full rank, orelse the psuedo-inverse of V if V is not of full rank.
 3. A statisticalprocess control method according to claim 2, in which a proximityindicator is used between the observation and an anomaly, referred to asan “angular proximity indicator”, obtained by measuring the anglebetween a vector representing the observation and a vector representingthe direction associate with the anomaly, the angular proximityindicator being the cosine of said angle, said cosine being calculatedby dividing the intensity indicator by the norm of the vectorrepresenting the observation, which norm is defined by said matrix V⁻¹.4. A statistical process control method according to claim 3, in which afunction F is used to define the control limit Lα of the angularproximity indicator beyond which an observation is allocated to saidanomaly, said function F representing a distribution function of thecosine of the angle (x) between an arbitrary given direction of R^(n),where n≧2, and a centered Gaussian random vector R^(n) having theintensity matrix as its covariance matrix, said distribution function Fof said random variable being given by the following formula:$\begin{matrix}{{{if}\quad x} < {- 1}} & {{F(x)} = 0} \\{{{if}\quad x} > 1} & {{F(x)} = 1} \\{{{if}\quad x} \geq {{- 1}\quad {and}\quad 1}} & {{F(x)} = {1 - \frac{S\left( {{ArcCos}(x)} \right.}{S(\pi)}}}\end{matrix}$where  S(θ) = ∫₀^(θ)sin^(n − 2)(t)  t  such  that    L  α = F⁻¹(α),  

where α represents the acceptable probability for a false alarm, αpreferably representing a value lying in the range 0.001% to 0.05%.
 5. Astatistical process control method according to claim 1, organized as: alearning stage which analyzes a history of the process to contribute toidentifying anomalies by performing the following steps: A1; priortransformation of the data; A2; separating measurements under controlfrom measurements out of control; A3; identifying the types of specialcauses and the associated direction parameters by a method ofautomatically classifying observations that is adapted to identifyobservations that are likely to be associated with the same anomaly, andthat enables observation points situated close to the same straight linecoming from the distribution center of the observations under control tobe grouped together; A4; creating and examining control charts on thespecial cause indicators; and A5; interpreting the special causes; acontrol stage proper which, for each new observation of the processvariables, diagnoses whether the observation depends on an anomaly, andif so which anomaly(ies) is/are probable, said stage comprising thefollowing three steps: S1; prior to transformation of the data; S2;detecting and identifying anomalies by inspecting control charts; andS3; where necessary identifying a combination of causes or a new specialcause and integrating that in the method.
 6. A statistical processcontrol method according to claim 3, in which the automaticclassification method used is of the hierarchical ascendingclassification type accepting the absolute value of the cosine as asimilarity index between two observation vectors and accepting themaximum binding criterion also known as the diameter criterion fordirecting the making up of groups.
 7. A statistical process controlmethod according to claim 5, in which the direction used in step A3 ofthe learning stage to represent an anomaly is the first inertial axis ofthe observation points associated with said anomaly, which axis passesthrough the distribution center of the under-control observations, saidaxis being determined by non-centered and non-reduced principalcomponent analysis on the cloud of observation points associated withthe anomaly, points previously centered on the distribution center ofthe under-control observation points.
 8. A statistical process controlmethod according to claim 5, in which control charts are used havingprobability thresholds with a logarithmic scale, the unilateral chartP_(u) being used when control relates to a single limit (as in the T²chart), the bilateral chart P_(b) being used when control relates to alower limit and an upper limit, said charts providing easierinterpretation and greater uniformity of presentation for chartsassociated with various probability distributions.